The Handbook in Monte Carlo Simulation Applications in Financial Engineering, Risk Management, and Economics features: An introductory section for basic material on stochastic modeling and estimation aimed at readers who may need a summary or review of the essentials Carefully crafted examples in order to spot potential pitfalls and drawbacks of each approach An accessible treatment of advanced topics such as low discrepancy sequences, stochastic optimization, dynamic programming, risk measures, and Markov chain Monte Carlo methods Numerous pieces of R code used to illustrate fundamental ideas in concrete terms and encourage experimentation
Preface xiii Part I Overview and Motivation 1 Introduction to Monte Carlo Methods 3 1.1 Historical origin of Monte Carlo simulation 4 1.2 Monte Carlo Simulation vs. Monte Carlo Sampling 7 1.3 System dynamics and the mechanics of Monte Carlo simulation 10 1.4 Simulation and optimization 21 1.5 Pitfalls in Monte Carlo simulation 30 1.6 Software tools for Monte Carlo simulation 35 1.7 Prerequisites 37 For further reading 38 Chapter References 38 2 Numerical Integration Methods 41 2.1 Classical quadrature formulae 43 2.2 Gaussian quadrature 48 2.3 Extension to higher dimensions: Product rules 53 2.4 Alternative approaches for high-dimensional integration 55 2.5 Relationship with moment matching 67 2.6 Numerical integration in R 69 For further reading 71 Chapter References 71 Part II Input Analysis: Modeling and Estimation 3 Stochastic Modeling in Finance and Economics 75 3.1 Introductory examples 77 3.2 Some common probability distributions 86 3.3 Multivariate distributions: Covariance and correlation 111 3.4 Modeling dependence with copulae 127 3.5 Linear regression models: a probabilistic view 136 3.6 Time series models 137 3.7 Stochastic differential equations 158 3.8 Dimensionality reduction 177 S3.1 Risk-neutral derivative pricing 190 S3.1.1 Option pricing in the binomial model 192 S3.1.2 A continuous-time model for option pricing: The Black--Scholes--Merton formula 194 S3.1.3 Option pricing in incomplete markets 199 For further reading 202 Chapter References 203 4 Estimation and Fitting 205 4.1 Basic inferential statistics in R 207 4.2 Parameter estimation 215 4.3 Checking the fit of hypothetical distributions 224 4.4 Estimation of linear regression models by ordinary least squares 229 4.5 Fitting time series models 232 4.6 Subjective probability: the Bayesian view 235 For further reading 244 Chapter References 245 Part III Sampling and Path Generation 5 Random Variate Generation 249 5.1 The structure of a Monte Carlo simulation 250 5.2 Generating pseudo-random numbers 252 5.3 The inverse transform method 263 5.4 The acceptance--rejection method 265 5.5 Generating normal variates 269 5.6 Other ad hoc methods 274 5.7 Sampling from copulae 276 For further reading 277 Chapter References 279 6 Sample Path Generation for Continuous-Time Models 281 6.1 Issues in path generation 282 6.2 Simulating geometric Brownian motion 287 6.3 Sample paths of short-term interest rates 298 6.4 Dealing with stochastic volatility 306 6.5 Dealing with jumps 308 For further reading 310 Chapter References 311 Part IV Output Analysis and Efficiency Improvement 7 Output Analysis 315 7.1 Pitfalls in output analysis 317 7.2 Setting the number of replications 323 7.3 A world beyond averages 325 7.4 Good and bad news 327 For further reading 327 Chapter References 328 8 Variance Reduction Methods 329 8.1 Antithetic sampling 330 8.2 Common random numbers 336 8.3 Control variates 337 8.4 Conditional Monte Carlo 341 8.5 Stratified sampling 344 8.6 Importance sampling 350 For further reading 363 Chapter References 363 9 Low-Discrepancy Sequences 365 9.1 Low-discrepancy sequences 366 9.2 Halton sequences 367 9.3 Sobol low-discrepancy sequences 374 9.4 Randomized and scrambled low-discrepancy sequences 379 9.5 Sample path generation with low-discrepancy sequences 381 For further reading 385 Chapter References 385 Part V Miscellaneous Applications 10 Optimization 389 10.1 Classification of optimization problems 390 10.2 Optimization model building 405 10.3 Monte Carlo methods for global optimization 412 10.4 Direct search and simulation-based optimization methods 416 10.5 Stochastic programming models 420 10.6 Scenario generation and Monte Carlo methods for stochastic programming 428 10.7 Stochastic dynamic programming 433 10.8 Numerical dynamic programming 440 10.9 Approximate dynamic programming 451 For further reading 453 Chapter References 453 11 Option Pricing 455 11.1 European-style multidimensional options in the BSM world 456 11.2 European-style path-dependent options in the BSM world 462 11.3 Pricing options with early exercise features 475 11.4 A look outside the BSM world 487 11.5 Pricing interest-rate derivatives 490 For further reading 497 Chapter References 498 12 Sensitivity Estimation 501 12.1 Estimating option greeks by finite differences 503 12.2 Estimating option greeks by pathwise derivatives 509 12.3 Estimating option greeks by the likelihood ratio method 513 For further reading 517 Chapter References 518 13 Risk Measurement and Management 519 13.1 What is a risk measure? 520 13.2 Quantile-based risk measures: value at risk 522 13.3 Monte Carlo methods for V@R 533 13.4 Mean-risk models in stochastic programming 537 13.5 Simulating delta-hedging strategies 540 13.6 The interplay of financial and nonfinancial risks 546 For further reading 548 Chapter References 548 14 Markov Chain Monte Carlo and Bayesian Statistics 551 14.1 An introduction to Markov chains 552 14.2 The Metropolis--Hastings algorithm 555 14.3 A re-examination of simulated annealing 558 For further reading 560 Chapter References 561 Index 563
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